208 IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 2, FEBRUARY 2011 Performance Analysis of Imperfect Channel Estimation in MIMO Two Hop Fixed Gain Relay Network with Beamforming Nuwan S. Ferdinand and Nandana Rajatheva, Senior Member, IEEE Abstract—We present the performance analysis of a MIMO two hop beamforming amplify-and-forward (AF) fixed gain relay system with imperfect channel estimation. Imperfect channel state information(CSI) will degrade the performance of the system. Here we quantify this effect by deriving the exact outage probability and the symbol error rate(SER). To gain more insight, we also present an asymptotic analysis which provides the details of the diversity order. Index Terms—Amplify-and-forward (AF), outage probability, symbol error rate (SER). I. I NTRODUCTION T HE performance analysis of an AF relay system with maximum ratio transmission(MRT) at source and maxi- mum ratio combining(MRC) at destination has been investi- gated previously in the literature. The effect of feedback delay has been investigated by the authors in [1] and the imperfect CSI at both the source and destination has been considered in [2]. The performance in both cases has been investigated for a CSI-assisted relay. In this paper we analyze a fixed gain relay system with imperfect CSI at both the source and the desti- nation which has not been reported previously to the best of our knowledge. Here both the source(S) and the destination(D) are equipped with multiple antennas and communicate via a single antenna relay(R). The S-R transmission has the form of MRT while MRC takes place at D. The MRT weight vector at the S is computed based on the outdated CSI and the MRC weight vector at the destination is computed based on CSI with Gaussian errors. To quantify the system performance due to imperfect CSI, we derive the closed form solution of outage probability and SER. Further, we analyze the system in high SNR which provides an insight into the performance and diversity order. II. SYSTEM MODEL We consider an AF two hop fixed gain relay system where S, equipped with T antennas and D having R antennas, com- municating via a single antenna R. Due to high shadowing we assume S does not have a direct link to D. The communication from S to D via the relay takes place in two time slots. In the first time slot, according to the transmit beamforming, S selects the MRT weight vector, w 1 (∣ ) and beamforms its Manuscript received September 21, 2010. The associate editor coordinating the review of this letter and approving it for publication was K. K. Wong. N. S. Ferdinand is with the field of ICT, School of Engineering and Tech- nology, Asian Institute of Technology P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand (e-mail: nuwanferdinand@gmail.com). N. Rajatheva is with the Centre for Wireless Communications, University of Oulu, Finland (e-mail: rajath@ait.ac.th, rrajathe@ee.oulu.fi). This research has been funded in part by the Academy of Finland grant 128010. Digital Object Identifier 10.1109/LCOMM.2011.010311.101770 signal, () with zero mean and unit variance to R. After the first time slot, the received signal(given in (1)) at the R is multiplied by a gain, , and then during the second time slot it is retransmitted to the D. ()= √ 1 w 1 (∣ )h 1 ()()+ 1 () (1) The received signal at is multiplied by the MRC weight vector, w 2 () and is given by, ()= w 2 () [ √ 2 ()h 2 ()+ n 2 () ] (2) 1 and 2 are the transmit power at S and R respectively, 1 () and n 2 () are 1 ×1 and R ×1 AWGN noise with variance 01 and 02 R respectively, where is the identity matrix of size . The MRT weight vector w 1 (∣ ) is computed based on the outdated CSI and is given by [3],w 1 (∣ )= h † 1 (− ) ∣h1(− )∣ , where (⋅) † denotes the Hermitian transpose. T × 1 vector, h 1 ( − ) is the time delayed channel gain vector of the perfect estimate h 1 () of S − R channel with Rayleigh fading entries. According to [3]–[5] the relationship between h 1 ( − ) and h 1 () can be expressed as, h 1 ()= h 1 ( − )+ e (), where = 0 (2 ) is the normalized correlation co- efficient between the h 1, () and h 1, ( − ); =1, .. T . 0 (⋅) is the zeroth oder Bessel function of the first kind, is the Doppler frequency and e () is the zero mean Gaussian error vector with variance (1 − 2 ) T . The MRC weight vector, w 2 ()= ˆ h † 2 () ∣ ˆ h2()∣ has Gaussian errors , where R × 1 vector ˆ h 2 () is the imperfect channel estimate of h 2 (), which has Rayleigh fading entries. In our analysis, according to [6, Eq.47], we assume that h 2 ()= ˆ h 2 ()+ e () where is the correlation parameter between h 2 () and ˆ h 2 () and e () is the zero mean Gaussian error vector with variance (1 − 2 ) R . The end to end SNR( ) can be obtained from (2) with some mathematical manipulation as, = 1 2 2 + . (3) where 1 = ∣w 1 (∣ )h 1 ()∣ 2 1 , 2 = ∣w 2 (∣ )h 2 ()∣ 2 2 , with 1 and 2 equal to 1 / 01 and 2 / 02 respectively. Fixed gain = 1 2 01 . The pdf of the random variable (RV), 1 can be found from [1, Eq.12], and the cdf can be obtained from, 1 (Λ) = ∫ Λ 0 1 ( ) . The pdf of 2 can be found from [6, Eq.46]. We select the as in [7, Eq.8], then the constant is given by = ( 1 [ 1 ( 1 + 1) ]) −1 (4) 1089-7798/11$25.00 c ⃝ 2011 IEEE